Triangulation and the Hauptvermutung

Related papers in the Homology Manifolds directory.

A triangulation of a topological space is a homeomorphism to simplicial complex.
The Hauptvermutung is the conjecture that any two triangulations of a topological space are combinatorially equivalent.
  • Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten by H. Tietze,
    Monatshefte fuer Math. und Phys. 19, 1-118 (1908).
    English translation (2008) by John Stillwell.
  • Beitraege zur Analysis situs by E. Steinitz, Sitz.-Ber. Berl. Math. Ges. 7, 29-49 (1908)
  • Die Topologie der Mannigfaltigkeiten by H. Kneser, Jahresb. D.M.V. 34, 1-13 (1926)
  • Page 152 of Topologie by P. Alexandroff and H. Hopf
    Springer (1935)
  • A proof of the generalized Schoenflies theorem by M.Brown, Bull. A.M.S. 66, 74--76 (1960)
  • Two complexes which are homeomorphic but combinatorially distinct by J.Milnor, Ann. of Math. 74, 575--590 (1961)
  • Microbundles I. by J.Milnor, Topology 3, suppl. 1, 53--80 (1964)
  • Microbundles are fibre bundles by J.Kister, Ann. of Math. 80, 190--199 (1964)
  • La classification des immersions combinatoires by A.Haefliger and V.Poenaru, Pub. Math. I.H.E.S. 23, 75-91 (1964)
  • Triangulating homotopy equivalences, Princeton Ph.D. thesis of D.Sullivan (1965)
  • On the Hauptvermutung for manifolds, by D.Sullivan, Bull. A.M.S. 73, 598--600 (1967)
  • The Hauptvermutung Book (edited by A.R.) K-Monographs in Mathematics 1, Springer (ex-Kluwer) (1996)
    Mathematical Reviews review Zentralblatt review
    Individual papers:
    On the Hauptvermutung by A.Ranicki (1996)
    Generalisations and Applications of Block Bundles by A.Casson (1967)
    Triangulating Homotopy Equivalences and Homeomorphisms by D.Sullivan (1967)
    The Princeton Notes on the Hauptvermutung by M.Armstrong, G.Cooke and C.Rourke (1968)
  • Stable homeomorphisms and the annulus conjecture, by R.Kirby, Ann. of Maths. (2) 89, 575--582 (1969)
  • On the triangulation of manifolds and the Hauptvermutung by R.Kirby and L.Siebenmann, Bull. A.M.S. 75, 742--749 (1968)
  • Counting topological manifolds by J.Cheeger and J.Kister, Topology 9. 149--151 (1970)
  • Topological Manifolds by L.Siebenmann, Proc. 1970 Nice ICM, Gauthier--Villars, Vol. 2, 133--163 (1971)
  • Some Recent Results on Topological Manifolds by R.Schultz, American Mathematical Monthly 78, 941--952 (1971)
  • Geometric periodicity and the invariants of manifolds by D.Sullivan, in Manifolds--Amsterdam 1970 (Proc. Nuffic Summer School)
    Springer Lecture Notes in Mathematics, Vol. 197, 44--75 (1971)
  • Deformations of spaces of imbeddings, by R.Edwards and R.Kirby, Annals of Maths. (2) 93, 63-88 (1971)
  • Recent Advances in Topological Manifolds, Notes of Part III Lecture course by Andrew Casson, Cambridge (1971).
    LATEX version
  • The canonical Schoenflies theorem by D.B.Gauld, Proc. A.M.S.27, 603--612 (1971)
  • Approximating cellular maps by homeomorphisms by L.Siebenmann, Topology 11, 271--294 (1972)
  • The triangulation of 3-manifolds by A.J.S.Hamilton, Quart. J. Math. Oxford Ser. (2) 27, 63--70 (1974)
  • Foundational Essays on Topological Manifolds, Smoothings, and Triangulations by R.Kirby and L.Siebenmann,
    Annals of Mathematics Studies 88, Princeton (1976). MR review.
    Maybe I'm doin' it wrong (MP3 file) Randy Newman song referred to on page 283. Lyrics.
  • Varietes simpliciales d'homologie et varietes topologiques metrisables by T.Matumoto, Orsay thesis (1976).
    Triangulation of manifolds, Proc. Symp. Pure Maths. Vol. 32 Part 2, 3--6, AMS (1978)
  • Classification of simplicial triangulations of topological manifolds by D.E.Galewski and R.J.Stern, Bull. AMS 82, 916--918 (1976)
    Simplicial triangulations of topological manifolds, Proc. Symp. Pure Maths. Vol. 32 Part 2, 7--12, AMS (1978)
  • The recognition problem: what is a topological manifold? by J.W.Cannon, Bull. A.M.S. 84, 832--866 (1978)
  • Hauptvermutung by H.Samelson, American Mathematical Monthly 85, 567--569 (1978)
  • A geometric proof of Rochlin's theorem by M.Freedman and R.Kirby, Proc. Symp. Pure Math. XXXII, Part 2, 85--97, AMS (1978)
  • Classification of simplicial triangulations of topological manifolds by D.E.Galewski and R.J.Stern, Ann. of Maths. (2) 111, 1--34 (1980)
  • The topology of manifolds and cell-like maps by R.D.Edwards, Proc. 1978 ICM Helsinki, 111--127 (1980)
  • The characterization of topological manifolds of dimension n≥5 by J.W.Cannon, Proc. 1978 ICM Helsinki, 449--454 (1980)
  • Algebraic L-theory and topological manifolds by A.Ranicki, Cambridge Tracts in Mathematics 102, CUP (1992)
  • Epsilon surgery by S.Ferry and E.K.Pedersen, Proc. 1993 Oberwolfach conference on the Novikov conjecture, Vol. 2,
    LMS Lecture Notes 227, CUP, 167--226 (1995)
    Section 13 is particularly relevant.
  • Some history of the Hauptvermutng (F.Quinn and A.R.) Letter to the Editor, Mathematical Intelligencer 19 (4), 5 (1997)
  • On Siebenmann periodicity by S.Hutt, Pacific J. Math. 183, 291--303 (1998)
  • Classical and Modern Topology by S.P. Novikov (2000). MR review.
  • Piecewise linear structures on topological manifolds by Y. Rudyak.
    e-print http://arxiv.org/abs/math.AT/0105047 (2001/2014).
  • Fragments of geometric topology from the sixties. by S. Buoncristiano, Geometry and Topology Monographs 6 (2003)
  • High dimensional manifold topology then and now Slides of 3 lectures by A.Ranicki at Siebenmann conference, Orsay (2005)
  • Oberwolfach Mini-Workshop on the Hauptvermutung for High-Dimensional Manifolds 13-19 August, 2006. Photo Report
  • A controlled-topology proof of the product structure theorem by F.Quinn, e-print GT/061031 (2006)
  • Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture, by Ciprian Manolescu. e-print http://arxiv.org/abs/1303.2354 (2013)
  • Aspherical manifolds that cannot be triangulated by Mike Davis, Jim Fowler and Jean-Francois Lafont. e-print http://arxiv.org/abs/1304.3730 (2013).
  • Non-triangulable manifolds Video of lecture by Rob Kirby on 24 May 2013, at the Bonn Arbeitstagung.
  • Triangulation of manifolds by Frank Quinn. e-print http://arxiv.org/abs/1310.7644 (2014).
  • Google for the Hauptvermutung